Characterization of singular graphs by order

Characterize all simple graphs of order n whose adjacency matrices have positive nullity, equivalently, determine all simple graphs of order n with singular adjacency matrix.

Background

The paper recalls a problem posed by Collatz and collaborators in 1957 concerning the characterization of graphs with nonzero adjacency-matrix nullity. Positive nullity is equivalent to adjacency-matrix singularity and has a chemical motivation: nullity zero is a necessary condition for the stability of certain conjugated molecules represented by carbon-skeleton graphs. The authors state that this characterization problem remains unresolved, although subsequent research has obtained bounds on rank or nullity in terms of graph parameters such as matching number, pendant vertices, maximum degree, and girth.

References

In 1957, Collatz et al. first wanted to characterize all graphs of order $n$ with $\eta(G)>0$. This question has strong chemical background, because $\eta(G)=0$ is a necessary condition for a so-called conjugated molecule to be chemically stable, where $G$ is the graph representing the carbon-atom skeleton of this molecule. Up to the present day, the problem has not been solved.

— The row left rank of a quaternion unit gain graph in terms of maximum degree  (2504.06674 - Lu et al., 9 Apr 2025) in Section 1, Introduction